Visualization of exponential fungal or bacterial growth. An existing colony grows and spawns new colonies nearby at exponentially increasing speed. Such a double exponential growth process leads to a powerlaw in the colony sizes. A stochastic microscopic model with the same macroscopic behavior is realized in the so called Chinese restaurant process, where new customers either select a random table with a likelihood proportional to the number of people already sitting at a table (colony growth) or opt to go to a new empty table (colony spawn). Here a new customer represents a newly created specimen in the dish. This type of preferential attachment is also known as Pareto principle, or the rich get richer, a simple universal mechanism to create broad scale free distributions.

Doubly exponential growth requires unlimited resources and unstinted growth everywhere in the dish. In reality, growth often only occurs on the border of the colonies and invasion of colony space may only happen by a new species occupying a different niche, i.e. requiring different or fewer resources or feeding on the other species. The distribution of colony sizes is not a powerlaw in this case (sublinear preferential attachment)